Lect6 - Stochastic Process Lecture 6 Fundamentals of...

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Stochastic Process 11/03/2006 Lecture 6 Fundamentals of Estimation II NCTUEE Summary In this lecture, I will discuss: Interval Estimator T Random Variable Maximum Likelihood Estimation Notation We will use the following notation rules, unless otherwise noted, to represent symbols during this course. Boldface upper case letter to represent MATRIX Boldface lower case letter to represent vector Superscript ( · ) T and ( · ) H to denote transpose and hermitian (conjugate transpose), respectively Upper case italic letter to represent RANDOM VARIABLE 6-1
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1 Confidence Interval (1) For an interval estimator [ L ( x ) , U ( x )] of a parameter θ based on the observation x , we say that the confidence coefficient of this interval is 1 - α if P h θ [ L ( x ) , U ( x )] i 1 - α, or we say [ L ( x ) , U ( x )] is a (1 - α ) × 100% confidence interval if P h L ( x ) θ U ( x ) i = 1 - α. Note: The random quantity here is the interval (based on the obser- vation x ), not the parameter θ . That is, the probability statements P £ L ( x ) θ U ( x ) / refers to x , not θ . Specifically, to find the proba- bility, we actually need to find P h L ( x ) θ U ( x ) i = P h x : L ( x ) θ and θ U ( x ) i . (2) Let’s see how to specify a confidence interval of the mean μ for two cases. (a) Unknown mean, known variance Let X 1 , · · · , X n be i.i.d. Gaussian variables with unknown mean μ and known variance σ 2 . The sample mean is a Gaussian random variable with ¯ X n N ( μ, σ 2 /n ) and ¯ X n - μ σ/ n N (0 , 1) . We can specify an interval [ - z, z ] within which the normalized sample mean has a probability P - z ¯ X n - μ σ/ n z = Q ( - z ) - Q ( z ) = 1 - 2 Q ( z ) , where Q ( z ) = R z 1 2 π e - y 2 / 2 dy is the standard Q-function. With simple algebraic efforts, the above can be rewritten as P ¯ X n - σz n μ ¯ X n + σz n = 1 - 2 Q ( z ) . (1) This means the interval [ ¯ X n - σz n , ¯ X n + σz n ] contains μ with proba- bility 1 - 2 Q ( z ). By letting α = 2 Q ( z ), we can find a corresponding z α/ 2 such that this interval is a (1 - α ) × 100% confidence interval for μ . 6-2
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(b) Unknown mean and unknown variance Let X 1 , · · · , X n be i.i.d. Gaussian variables with unknown mean μ and unknown variance σ 2 . The confidence interval now becomes [ ¯ X n - S n z n , ¯ X n + S n z n ] , where the variance σ 2 is replaced by the sample variance S 2 n . So, the probability of μ containing in this interval is P ¯ X n - S n z n μ ¯ X n + S n z n = P - z ¯ X n - μ S n / n | {z } , T z . The random variable involved in figuring out the above probability measure is T , ¯ X n - μ S n / n . We need to find the pdf of T in order to specify the interval. The random variable T is called Student’s T random variable. With some rearrangement, we see T = ¯ X n - μ S n / n = ¯ X n - μ σ/ n q ( n - 1) S 2 n σ 2 ( n - 1) , where the numerator ¯ X n - μ σ/ n is a standard normal random variable independent with ( n - 1) S 2 n σ 2 , which is a chi-squared random variable with n - 1 degree of freedom, in the denominator.
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